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Nonlinear Elastic-Net Regularization and Its Iterative Soft Thresholding Algorithm

  • Yu Tian and Liang Ding EMAIL logo
Published/Copyright: November 10, 2025

Abstract

Elastic-net regularization, as a variational method, demonstrates enhanced stability compared to classical 1 sparsity regularization, making it suitable for addressing ill-conditioned problems. However, conventional elastic-net regularization is typically limited to linear equations. In this paper, we extend the elastic-net regularization method to nonlinear problems. We investigate the well-posedness of this regularization and demonstrate that it serves as a sparsity regularization approach. The iterative soft thresholding algorithm, commonly used for classical 1 sparsity regularization, features a straightforward structure and is easy to implement. We show that, under widely accepted conditions regarding the nonlinearity of the function F, this algorithm is effective in solving the elastic-net regularization for nonlinear ill-conditioned equations. Our numerical results highlight the efficiency of the proposed method.

MSC 2020: 49M37; 65K10

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Received: 2024-11-07
Revised: 2025-10-07
Accepted: 2025-10-21
Published Online: 2025-11-10

© 2025 Institute of Mathematics of the National Academy of Science of Belarus, published by De Gruyter, Berlin/Boston

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